2.4 farmer bought a chaise for 210 dollars, a horse for 70 dollars, and a saddle for 9 dollars; what was the whole amount? Write the numbers as before directed, with units under units, tens under tens, &c. OPERATION. Chaise, 210 dollars. Horse, 70 dollars. Answer, 289 dollars. Add as before. The units will be 9, the tens 8, and the hundreds 2; that is, 210-+70+9=289. After the same manner are performed the following examples, in which the amount of no column exceeds nine. 3. A man had 15 sheep in one pasture, 20 in another pasture, and 143 in another; how many sheep had he in the three pastures? 15+20+143= how many? 4. A man has three farms, one containing 500 acres, another 213 acres, and another 76 acres; how many acres in the three farms? 500+213 +76: how many? = 5. Bought a farm for 2316 dollars, and afterwards sold it so as to gain 550 dollars; what did I sell the farm for? 2316 +550 how many? = 6. A chair-maker sold, in one week, 30 Windsor chairs, 36 cottage, 102 fancy, and 21 Grecian chairs; how many chairs did he sell? 30+36 +102+21= how many? 7. A farmer, after selling 500 bushels of wheat to a commission merchant, 320 to a miller, and sowing 117 bushels, found he had 62 bushels left; how many bushels had he at first? 500+320+117+62= how many? 8. A dairyman carried to market at one time 231 pounds of butter, at another time 124, at another 302, at another 20, and at another 12; how many pounds did he carry in all? Ans. 689 pounds. 9. A box contains 115 arithmetics, 240 grammars, 311 geographies, 200 reading books, and 133 spelling books; how many books are there in the box? Ans. 999. ¶ 13. Hitherto the amount of any one column, when added up, has not exceeded, 9, and consequently has been expressed by a single figure. But it will frequently happen that the amount of a single column will exceed 9, requiring two or more figures to express it. 1. There are three bags of money. The first contains 876 tollars, the second 653 dollars, the third 426 dollars; what is the amount contained in all the bags ? OPERATION. First bag, 876 dollars. SOLUTION. -Writing the numbers as already described, we add the units, and find them to be 15, equal to 5 units, which we write in units' place, adding the 1 ten with the tens; which being added together are 15 tens, equal to 5 tens, to be written in tens' place, and 1 hundred, to be added to the hundreds. The hundreds being added are 19, equal to 9 hundreds, to be written in hundreds' place, and 1 thousand, to be written in thousands' place. 1955 66 Ans. 1955 dollars. PROOF.-We may reverse the order, and, beginning at the top, add the figures downwards. If the two results are alike, the work may be supposed to be right, for it is not likely that the same mistake will be made twice, when the figures are added in a different order. NOTE.-Proof by the excess of nines. If the work be right, there will be just as many of any small number, as 9, with the same remainder, in the amount, as in the several numbers taken together. Hence, OPERATION. 876 3 653 5 426 3 In the upper number, 8 (hundreds) is 8 more than a certain number of nines, (5) 7 (tens) is 7 more. Adding the 8 and 7, and the 6 units together, the sum is 212 nines and 3 remainder, which we set down at the right hand, as the excess of nines in this number. In the same manner, 5 is found to be the excess of nines 1955 2 in the second number, and 3 in the third number. These several excesses being added together, make 1 nine and an excess of 2, which is the same as the excess of nines in the general amount, found in the same manner. This method will detect every mistake, except it be 9, or an exact number of nines. To find what will be the excess after casting the nines out of any number, begin at the left hand, and add together the figures which express the number; thus, to cast the nines out of 892, we say 8 (passing over 9)+2(dropping 9 from the sum) = 1. From the examples and illustrations now given, we derive the following RULE, 1. Write the numbers to be added, one under another, plac Questions. T 13. If the amount of the column does not exceed 9, what do you do? What when it exceeds 9? How do you add each column? What do you do with the amount of the left column? For what number do you carry? If the amount of a column be 36, what would you set down, and how many would you carry? On what principle do you do this? How is addition proved? Why? Repeat the rule for addition. ing units under units, tens under tens, &c., and draw a line underneath. II. Begin at the unit column, and add together all the figures contained in it; if the amount does not exceed 9, write it under the column; but if it exceed 9, write the units in units' place, and carry the tens to the column of tens. III. Add each succeeding column in the same manner, and set down the whole amount of the last column. 4. Add together 587, 9658, 67, 431, 28670, 85, 100000, 6300, and 1. Amount, 145799. 5. What is the amount of 8635, 7, 2194, 16, 7421, 93, 5063, 135, 2196, 89, and 1225? Ans. 27074. 6. A man being asked his age, answered that he left England when he was 12 years old, and that he had afterwards spent 5 years in Holland, 17 years in Germany, 9 years in France, whence he sailed for the United States in the year 1827, where he had lived 22 years; what was his age? Ans. 65 years. 7. A company contract to build six warehouses; for the first they receive 36850 dolls.; for the second, 43476 dolls.; for the third, 18964 dolls. ; for the fourth, 62840 dolls. ; for the fifth, 71500 dolls.; for the sixth, as much as for the first three; to what do these contracts amount? Ans. 332920 dollars. 8. James had 7 marbles, Peter had 4 marbles more than James, and John had 5 more than Peter; how many marbles in all? Ans. 34. 9. There are seven men; the first man is worth 67850 dollars; the second man is worth 2500 dolls. more than the first man; the third, 3168 dolls. more than the second; the fourth, 16973 dolls. more than the third; the fifth, 40600 dolls. more than the fourth; the sixth, 19888 dolls. more than the fifth; and the seventh, 49676 dolls. more than the sixth; how many dollars are they all worth? Ans. 784934 dollars. 10. What is the interval in years between a transaction that happened 275 years ago, and one that will happen 125 years hence? Ans. 400 years. 11. What is the amount of 46723, 6742, and 986 dollars? 12. A man has three orchards; in the first there are 14C trees that bear apples, and 64 trees that bear cherries; in the second, 234 trees bear apples, and 73 bear cherries; in the third, 47 trees bear plums, 36 bear pears, and 25 bear cherries; how many trees in all the orchards, and how many of each kind? Ans. 619 trees; 374 bear apples; 162 cherries; 47 plums; and 36 pears. 13. A gentleman purchased a farm for 7854 dollars; he paid 194 dollars for having it drained and fenced, and 300 dollars for having a barn built upon it; how much did it cost him, and for how much must he sell it, to gain 273 dollars? It cost him 8348 dollars. He must sell it for 8621 dollars. Ans. ¶ 14. Review of Numeration and Addition. Questions. What are numbers? What are the methods of expressing numbers? What is numeration? notation? fundamental law in the Arabic notation? How does the Arabic differ from the Roman method? What is understood by units of different orders? What is quantity? Arithmetic? What is understood by the simple value of figures? the local value? the unit value of a number? Explain the difference between an abstract and a denominate number. What is addition? the rule? proof? For what number do you carry, and why? EXERCISES. 1. Washington was born in the year of our Lord 1732; he was 67 years old when he died; in what year did he die? Ans. 1799. 2. The invasion of Greece by Xerxes took place 481 years before Christ; how long ago is that this current year? 3. There are two numbers; the less is 8671, the difference between the numbers is 597; what is the greater number? Ans. 9268. 4. A man borrowed a sum of money, and paid in part 684 dollars; the sum left unpaid was 876 dollars; what was the sum borrowed? 5. There are four numbers; the first 317, the second 812, the third 1350, and the fourth as much as the other three ; what is the sum of them all? Ans. 4958. 6. A gentleman left his daughter 16 thousand 16 hundred and 16 dollars; he left his son 1800 more than his daughter; what was his son's portion, and what was the amount of the whole estate? Son's portion, 19416. Ans. Whole estate, 37032. 7. A man, at his death, left his estate to his four children who, after paying debts to the amount of 1476 dollars, received 4768 dollars each; how much was the whole estate? Ans. 20548. 8. A man bought four hogs, each weighing 375 pounds; how much did they all weigh? Ans. 1500. 9. The fore quarters of an ox weigh one hundred and eight pounds each, the hind quarters weigh one hundred and twenty-four pounds each, the hide seventy-six pounds, and the tallow sixty pounds; what is the whole weight of the ox? Ans. 600. 10. The imports into the several States in 1842, were as follows: Me. 606864 dollars, N. H. 60481, Vt. 209868, Mass. 17986433, R. I. 323692, Ct. 335707, N. Y. 57875604, N. J. 145, Pa. 7385858, Del. 3557, Md. 4417078, D. C. 29056, Va. 316705, N. C. 187404, S. C. 1359465, Ga. 341764, Al. 363871, La. 8033590, O. 13051, Ky. 17306, Tenn. 5687, Mich. 80784, Mo. 31137, Fa. 176980 dollars; what was the entire amount? Ans. 100162087. SUBTRACTION OF SIMPLE NUMBERS. ¶ 15. 1. Charles, having 18 cents, bought a book, for which he gave 6 cents; how many cents had he left? 2. John had 12 apples; he gave 5 of them to his brother; how many had he left? 3. Peter played at marbles; he had 23 when he began, but when he had done he had only 12; how many did he lose? 4. A man bought a cow for 17 dollars, and sold her again for 22 dollars; how many dollars did he gain? 5. Charles is 9 years old, and Andrew is 13; what is the difference in their ages? 6. A man borrowed 50 dollars, and paid all but 18; how many dollars did he pay? that is, take 18 from 50, and how many would there be left? The taking of a less number from a greater (as in the foregoing examples) is called Subtraction. The greater number |